A mathematician is bound to be horrified by my mathematical comments, — Mathematics

"As soon as a thought or word becomes a tool, one can dispense with actually ‘thinking’ it, that is, with going through the logical acts involved in verbal formulation of it. As has been pointed out, often and correctly, the advantage of mathematics—the model of all neo-positivistic thinking—lies in just this ‘intellectual economy.’ Complicated logical operations are carried out without actual performance of the intellectual acts upon which the mathematical and logical symbols are based. … Reason … becomes a fetish, a magic entity that is accepted rather than intellectually experienced."

"Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat flour from peascods, so pages of formulæ will not get a definite result out of loose data."

"In doing mathematics, I express something personal. It is a source of joy to know that, despite this personal aspect, the fruit of my work can be of interest to other mathematicians."

"If a religion is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one."

"It might have been beans, successful hunts, or victories in battle, but, for millennia, people were using maths to describe things - counting them, measuring them, dividing them up."

"“There’s more to life than mathematics,” Joan said. “But not much more.”"

"I appeal to all pupils, students and young people, asking you to focus on the horizons that are opening up for you, and which you could only dream of a year ago. Our future will depend on your desire for education and moral values as well as on your entrepreneurial spirit."

"We have created a wealthy society with tens of millions of talented, resourceful individuals who play virtually no role whatsoever as citizens. Bringing these people in — with their networks of influence, their knowledge, and their resources — is the key to creating the capacity for shared intelligence that we need to solve our problems."

"We are shocked when we see educators, timid before criticism and confused about first principles, betray their trust. And we wonder what can be that philosophy of education which believes that young people can be trained to the duties of citizenship by wrapping their minds in cotton wool."

"An [hypertext] encyclopaedia will be an overall attempt by the knowledgeable, the learned societies or anyone else, to represent the state-of-the-art in their field. An encyclopaedia will be a living document, as up to date as it can be, instantly accessible at any time. It will contain carefully authored explanations and summaries of the subject, as well as computer-generated indexes of literature. A reference to a paper from the encyclopaedia conveys authority and acceptance by academic society. A measure of a paper’s standing may be conveyed by the number of links it is away from an encyclopaedia."

"A sample of the modern debate, which neatly summarizes the anti-reductionist position is provided by Grene (1974). She points out that in principle a one-level ontology—the belief, for example, that with increasing knowledge all science will become an account of the world in the language of, say, atomic events—contradicts itself. This is so because such a belief, to be meaningful, requires an ontology which admits both atomic events and cognition. Here at once a second level is smuggled in! It is logically possible... that there might be no levels in between those of atomic events and cognition (that in essence is Descartess position) but the sciences of chemistry and biology consist of some well-tested conjectures that there are such intermediate levels as are represented by molecules, cells, organelles, organs, and organisms."

"All competent thinkers agree with Bacon that there can be no real knowledge except that which rests upon observed facts. This fundamental maxim is evidently indisputable if it is applied, as it ought to be, to the mature state of our intelligence. But, if we consider the origin of our knowledge, it is no less certain that the primitive human mind could not, and indeed ought not to, have thought in that way. For if, on the one hand, every Positive theory must necessarily be founded upon observations, it is, on the other hand, no less true that, in order to observe, our mind has need of some theory or other. If in contemplating phenomena we did not immediately connect them with principles, not only would it be impossible for us to combine these isolated observations, and therefore to derive profit from them, but we should even be entirely incapable of remembering facts, which would for the most remain unnoted by us. Thus there were two difficulties be overcome: the human mind had to observe in order to form real theories, and yet had to form theories of some sort before it could apply itself to a connected series of observations. The primitive human mind, therefore, found itself involved in a vicious circle, from which it would never have had any means of escaping, if a natural way of the difficulty had not fortunately found by the spontaneous development of Theological conceptions. ...chimerical hopes ..exaggerated ideas of mans importance in the universe to which the Theological Philosophy ...at the commencement, ...afforded an indispensable stimulus without the aid which we cannot, indeed, conceive how the primitive human mind would have been induced to undertake any arduous labours."